There are two categories of methods widely used in NWP, namely variational methods and the ensemble Kalman filter (EnKF) [1]}, [2]}. The former is based on a weighted least-square optimization, such as the four dimensional variational data assimilation (4D-Var) in a fixed time window or the three dimensional version (3D-Var) that excludes the time variable. The EnKF algorithm is based on the Kalman filter except that the error covariance is approximated using a set of state ensembles. 4D-Var methods are used in operational NWP systems by many meteorological centers. While it serves as an effective method of data assimilation, 4D-Var algorithms have difficulty to explicitly track the evolution of error covariance within its estimation process due to outrageous computational costs and input/output (I/O) loads required by matrices of extremely high dimensions. EnKF, on the other hand, updates information about the error covariance in the form of ensembles. However, the number of ensemble states is significantly smaller than the number of state variables. As a result, the rank deficiency of error covariance tends to deteriorate the integrity of the estimation process unless remedies to the algorithm, such as localization and covariance inflation, are applied.
